Download Geometry Review Study Pack The following equations may be

Geometry Review Study Pack
The following equations may be useful for this work package. They are all IB questions starting ranging
in value from 6 points to 14 points.
Equations:
Distance between two points ( x1 , y1 ) and ( x2 , y2 ) : d 
( x1  x 2 ) 2  ( y1  y 2 )
 x1  x2 y1  y 2 
,

2 
 2
Coordinates of the midpoint of a line segment with endpoints ( x1 , y1 ) and ( x2 , y2 ) : 
Equation of a straight line: y  mx  c
Gradient Formula:
1.
ax  bx  d  0
y 2  y1
x2  x1
The diagram below shows the line with equation 3x + 2y = 18. The points A and B are the y and xintercepts respectively. M is the midpoint of [AB].
y
A
3x + 2y =18
M
Diagram not to scale
B
O
x
Find the coordinates of
(a)
the point A;
(b)
the point B;
c)
the point M.
(Total 8 marks)
2.
The mid-point, M, of the line joining A(s, 8) to B(−2, t) has coordinates M(2, 3).
(a)
Calculate the values of s and t.
(2)
(b)
Find the equation of the straight line perpendicular to AB, passing through
the point M.
(4)
(Total 6 marks)
3.
Three points A (1, 3), B (4, 10) and C (7, –1) are joined to form a triangle. The mid-point of AB is D and
the mid-point of AC is E.
(a)
Plot the points A, B, C, on the grid.
(b)
Find the distance DE
(Total 6 marks)
1
4. P (4, 1) and Q (0, –5) are points on the coordinate plane.
(a)
Determine the
(i)
coordinates of M, the midpoint of P and Q;
(ii)
gradient of the line drawn through P and Q;
(iii)
gradient of the line drawn through M, perpendicular to PQ.
The perpendicular line drawn through M meets the y-axis at R (0, k).
(b)
Find k.
(Total 6 marks)
5.
y
3
2
1
–5 –4 –3 –2 –1
–1
0
1
2
3
4
5
x
–2
–3
–4
–5
–6
(a)
On the grid above, draw a straight line with a gradient of –3 that passes through the point (–2, 0).
(b)
Find the equation of this line.
(Total 8 marks)
6.
Three points are given A(0, 4), B(6, 0) and C(8, 3).
(a)
Calculate the gradient (slope) of line AB.
(2)
(b)
Find the coordinates of the midpoint, M, of the line AC.
(2)
(c)
Calculate the length of line AC.
(2)
(d)
Find the equation of the line BM giving your answer in the form ax + by + d = 0 where a, b and d
 .
(5)
(e)
State whether the line AB is perpendicular to the line BC showing clearly your working and
reasoning.
(3)
(Total 14 marks)
2
7.
The coordinates of the vertices of a triangle are P (–2, 6), Q (6, 2) and R (–8, a).
(a)
On graph paper, mark the points P and Q on a set of coordinate axes.
Use 1 cm to represent 1 unit on each axis.
(3)
(b)
(i)
Calculate the distance PQ.
(2)
(ii)
Find the gradient of the line PQ.
(3)
(iii)
If angle RPQ is a right angle, what is the gradient of the line PR?
(1)
(iv)
Use your answer from (b) (iii), or otherwise, to find the value of ‘a’.
(2)
(c)
The length of PR is 180 . Find the area of triangle PQR.
(2)
(Total 13 marks)
8.
On the coordinate axes below, D is a point on the y-axis and E is a point on the x-axis.
O is the origin. The equation of the line DE is y 
y
1
x = 4.
2
(not to scale)
D
C
O
(a)
B
E
x
Write down the coordinates of point E.
(2)
C is a point on the line DE. B is a point on the x-axis such that BC is parallel to the y-axis. The xcoordinate of C is t.
(b)
1
Show that the y-coordinate of C is 4  t.
2
(2)
OBCD is a trapezium. The y-coordinate of point D is 4.
(c)
1
Show that the area of OBCD is 4t  t 2 .
4
(3)
(d)
The area of OBCD is 9.75 square units. Write down a quadratic equation that expresses this
information.
(1)
3
(e)
(i)
Using your graphic display calculator, or otherwise, find the two solutions to the quadratic
equation written in part (d).
(ii)
Hence find the correct value for t. Give a reason for your answer.
(4)
(Total 12 marks)
9.
1).
The vertices of quadrilateral ABCD as shown in the diagram are A (–8, 8), B (8, 3), C (7,–1) and D (–4,
y
A
8
7
6
5
4
3
B
2
1
D
–8
–7
–6
–5
–4
–3
–2
–1
0
1
2
3
4
5
6
–1
7
8
x
C
–2
The gradient of the line AB is –
(a)
5
.
16
Calculate the gradient of the line DC.
(2)
(b)
State whether or not DC is parallel to AB and give a reason for your answer.
(2)
The equation of the line through A and C is 3x + 5y = 16.
(c)
Find the equation of the line through B and D expressing your answer in the form ax + by = c,
where a, b and c  .
(5)
The lines AC and BD intersect at point T.
(d)
Calculate the coordinates of T.
(4)
(Total 13 marks)
4